{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# The Laplace Transform\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inverse Transform\n",
    "\n",
    "So far only the [(forward) Laplace transform](definition.ipynb) has been introduced. The Laplace transform features also an [inverse transform](https://en.wikipedia.org/wiki/Inverse_Laplace_transform). The inverse Laplace transform maps a complex-valued Laplace transform $X(s) \\in \\mathbb{C}$ with complex-valued independent variable $s \\in \\mathbb{C}$ into the complex-valued signal $x(t) \\in \\mathbb{C}$ with real-valued independent variable $t \\in \\mathbb{R}$. It can be shown that the inverse Laplace transform $x(t) = \\mathcal{L}^{-1} \\{ X(s) \\}$ is uniquely determined for most practically relevant signals. This section discusses two different techniques for the computation of the inverse Laplace transform."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integral Formula\n",
    "\n",
    "Using results from complex analysis, the inverse Laplace transform is given by the following complex line integral\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = \\frac{1}{2 \\pi j} \\int_{\\sigma - j \\infty}^{\\sigma + j \\infty} X(s) \\, e^{s t} \\; ds\n",
    "\\end{equation}\n",
    "\n",
    "where $X(s) = \\mathcal{L} \\{ x(t) \\}$ is assumed to be analytic in its simply connected region of convergence (ROC). The notation $\\sigma \\mp j \\infty$ for the lower/upper integration limit denotes an arbitrary integration path which lies in the ROC and ranges from $\\Im \\{s\\} = - \\infty$ to $\\Im \\{s\\} = + \\infty$. The integration path can be chosen parallel to the imaginary axis but also all other paths in the ROC are possible. This results from [Cauchy's integral theorem](https://en.wikipedia.org/wiki/Cauchy's_integral_theorem). Two equivalent paths are shown in the following illustration\n",
    "\n",
    "![Possible integration paths for the inverse Laplace transform](integration_paths.png)\n",
    "\n",
    "where the blue line indicates the integration path and the gray area the ROC."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rational Laplace Transforms\n",
    "\n",
    "Computing the inverse Laplace transform by above integral formula can be challenging. The [Cauchy residue theorem](https://en.wikipedia.org/wiki/Residue_theorem) provides a practically tractable solution for Laplace transforms $X(s) = \\mathcal{L} \\{ x(t) \\}$ which are given as rational functions. It states that the value of a line integral of an holomorphic function over a closed contour is given by summing up its [residues](https://en.wikipedia.org/wiki/Residue_theorem). The residue is the value of the line integral for a path enclosing a singularity. Consequently, the inverse Laplace transform of a rational Laplace transform can be computed by summing up the individual contributions from its poles. This procedure is detailed in the following."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Basic Procedure\n",
    "\n",
    "A rational Laplace transform $X(s)$ can be written in terms of its numerator and denominator polynomial\n",
    "\n",
    "\\begin{equation}\n",
    "X(s) = \\frac{\\sum_{m=0}^{M} \\beta_m s^m}{\\sum_{n=0}^{N} \\alpha_n s^n}\n",
    "\\end{equation}\n",
    "\n",
    "where $M$, $N$ denote the order of the numerator and denominator polynomial and $\\beta_m$, $\\alpha_n$ their coefficients, respectively. It is assumed that $\\alpha_N \\neq 0$ and that $M \\leq N$. If $M > N$, $X(s)$ can be decomposed by [polynomial division](https://en.wikipedia.org/wiki/Polynomial_long_division) into a sum of powers of $s$ and a rational function fulfilling $M \\leq N$. \n",
    "\n",
    "Now a [partial fraction decomposition](https://en.wikipedia.org/wiki/Partial_fraction_decomposition) of $X(s)$ is performed resulting in\n",
    "\n",
    "\\begin{equation}\n",
    "X(s) = A_0 + \\sum_{\\mu = 1}^{L} \\sum_{\\nu = 1}^{R_\\mu} \\frac{A_{\\mu \\nu}}{(s - s_{\\infty \\mu})^\\nu}\n",
    "\\end{equation}\n",
    "\n",
    "where $s_{\\infty \\mu}$ denotes the $\\mu$-th unique pole of $X(s)$, $R_\\mu$ its degree and $L$ the total number of different poles $\\mu = 1 \\dots L$. Using the known Laplace transforms (cf. [example for the modulation theorem](theorems.ipynb#Modulation-Theorem) or [table of selected transforms](table_theorems_transforms.ipynb#Selected-Transforms))\n",
    "\n",
    "\\begin{equation}\n",
    "\\mathcal{L} \\{ t^n e^{-s_0 t} \\epsilon(t) \\} = \\frac{n!}{(s + s_0)^{n+1}} \\qquad \\text{for } \\Re \\{ s \\} > \\Re \\{ - s_0 \\}\n",
    "\\end{equation}\n",
    "\n",
    "and $\\mathcal{L} \\{ \\delta(t) \\} = 1$, together with the linearity of the Laplace transform yields a generic result for the inverse Laplace transform $x(t) = \\mathcal{L}^{-1} \\{ X(s) \\}$ of a right-sided signal\n",
    "\n",
    "\\begin{equation}\n",
    "x(t) = A_0 \\cdot \\delta(t) + \\epsilon(t) \\sum_{\\mu = 1}^{L} e^{s_{\\infty \\mu} t} \\sum_{\\nu = 1}^{R_\\mu} \\frac{A_{\\mu \\nu} \\, t^{\\mu - 1}}{(\\nu -1)!}\n",
    "\\end{equation}\n",
    "\n",
    "It remains to compute the coefficients $A_0$ and $A_{\\mu \\nu}$ of the partial fraction decomposition. The constant coefficient $A_0$ is given as\n",
    "\n",
    "\\begin{equation}\n",
    "A_0 = \\lim_{s \\to \\infty} X(s)\n",
    "\\end{equation}\n",
    "\n",
    "For a pole $s_{\\infty \\mu}$ with degree $R_\\mu = 1$, the coefficient $A_{\\mu 1}$ reads\n",
    "\n",
    "\\begin{equation}\n",
    "A_{\\mu 1} = \\lim_{s \\to s_{\\infty \\mu}} \\left( X(s) \\cdot (s - s_{\\infty \\mu}) \\right)\n",
    "\\end{equation}\n",
    "\n",
    "For a pole $s_{\\infty \\mu}$ of degree $R_\\mu > 1$, the coefficients $A_{\\mu \\nu}$ are given as\n",
    "\n",
    "\\begin{equation}\n",
    "A_{\\mu \\nu} = \\frac{1}{(R_\\mu - \\nu)!} \\lim_{s \\to s_{\\infty \\mu}} \\frac{d^{R_\\mu - \\nu}}{d s^{R_\\mu - \\nu}} \\left( X(s) \\cdot (s - s_{\\infty \\mu})^{R_\\mu} \\right)\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Inverse transform of a rational Laplace transform\n",
    "\n",
    "The inverse transform $x(t) = \\mathcal{L}^{-1} \\{ X(s) \\}$ of \n",
    "\n",
    "\\begin{equation}\n",
    "X(s) = \\frac{1}{(s+1) (s+2)^2} \\qquad \\text{for } \\Re \\{s \\} > -1\n",
    "\\end{equation}\n",
    "\n",
    "is computed using the procedure outline above. First the function $X(s)$ is defined in `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{1}{\\left(s + 1\\right) \\left(s + 2\\right)^{2}}$$"
      ],
      "text/plain": [
       "       1        \n",
       "────────────────\n",
       "               2\n",
       "(s + 1)⋅(s + 2) "
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "%matplotlib inline\n",
    "sym.init_printing()\n",
    "\n",
    "s = sym.symbols('s', complex=True)\n",
    "t = sym.symbols('t', real=True)\n",
    "\n",
    "X = 1/((s+1)*(s+2)**2)\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since $X(s)$ has two real-values poles, $x_{\\infty 1} = -1$ with degree $R_1 = 1$ and $x_{\\infty 2} = -2$ with degree $R_2 = 2$, the following partial fraction decomposition is chosen as ansatz in accordance with above given formula"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$A_{0} + \\frac{A_{11}}{s + 1} + \\frac{A_{21}}{s + 2} + \\frac{A_{22}}{\\left(s + 2\\right)^{2}}$$"
      ],
      "text/plain": [
       "     A_{11}   A_{21}    A_{22} \n",
       "A₀ + ────── + ────── + ────────\n",
       "     s + 1    s + 2           2\n",
       "                       (s + 2) "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A0, A11, A21, A22 = sym.symbols('A_0 A_{11} A_{21} A_{22}', real=True)\n",
    "Xp = A0 + A11/(s+1) + A21/(s+2) + A22/(s+2)**2\n",
    "Xp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The four real-valued constants $A_0$, $A_{11}$, $A_{21}$ and $A_{22}$ of the partial fraction decomposition will be determined later. First a look is taken at the inverse Laplace transforms of the individual summands composed from the constant and the individual poles. For the constant we get"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$A_{0} \\delta\\left(t\\right)$$"
      ],
      "text/plain": [
       "A₀⋅δ(t)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = sym.inverse_laplace_transform(Xp.args[0], s, t)\n",
    "x0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the first pole $s_{\\infty 1}$ with degree zero we get"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$A_{11} e^{- t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "        -t             \n",
       "A_{11}⋅ℯ  ⋅Heaviside(t)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x1 = sym.inverse_laplace_transform(Xp.args[1], s, t)\n",
    "x1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the second pole $s_{\\infty 2}$ with degree two we get two contributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$A_{21} e^{- 2 t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "        -2⋅t             \n",
       "A_{21}⋅ℯ    ⋅Heaviside(t)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x2 = sym.inverse_laplace_transform(Xp.args[2], s, t)\n",
    "x2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$A_{22} t e^{- 2 t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "          -2⋅t             \n",
       "A_{22}⋅t⋅ℯ    ⋅Heaviside(t)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x3 = sym.inverse_laplace_transform(Xp.args[3], s, t)\n",
    "x3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inverse Laplace transform of $X(s)$ is now composed from the superposition of the four individual parts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$A_{0} \\delta\\left(t\\right) + A_{11} e^{- t} \\theta\\left(t\\right) + A_{21} e^{- 2 t} \\theta\\left(t\\right) + A_{22} t e^{- 2 t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "                  -t                        -2⋅t                          -2⋅t\n",
       "A₀⋅δ(t) + A_{11}⋅ℯ  ⋅Heaviside(t) + A_{21}⋅ℯ    ⋅Heaviside(t) + A_{22}⋅t⋅ℯ    \n",
       "\n",
       "             \n",
       "⋅Heaviside(t)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = x0 + x1 + x2 + x3\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the final solution the four coefficients are determined using the given limit formulas for the coefficients of a partial fraction decomposition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ A_{0} : 0, \\quad A_{11} : 1, \\quad A_{21} : -1, \\quad A_{22} : -1\\right \\}$$"
      ],
      "text/plain": [
       "{A₀: 0, A_{11}: 1, A_{21}: -1, A_{22}: -1}"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "coeffs = {A0 : sym.limit(X, s, sym.oo)}\n",
    "coeffs.update({A11 : sym.limit(X*(s+1), s, -1)})\n",
    "coeffs.update({A21 : sym.limit(sym.diff(X*(s+2)**2, s), s, -2)})\n",
    "coeffs.update({A22 : sym.limit(X*(s+2)**2, s, -2)})\n",
    "coeffs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substitution into the inverse Laplace transform yields the final solution $x(t) = \\mathcal{L}^{-1}\\{ X(s) \\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- t e^{- 2 t} \\theta\\left(t\\right) + e^{- t} \\theta\\left(t\\right) - e^{- 2 t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "     -2⋅t                 -t                 -2⋅t             \n",
       "- t⋅ℯ    ⋅Heaviside(t) + ℯ  ⋅Heaviside(t) - ℯ    ⋅Heaviside(t)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = x.subs(coeffs)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The solution is plotted for illustration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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   "source": [
    "sym.plot(x, (t, -1, 10), ylabel=r'$x(t)$')"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Derive the inverse Laplace transform of $X(s)$ by manual calculation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Classification of Poles\n",
    "\n",
    "Above procedure allows to compute the inverse Laplace transform $x(t) = \\mathcal{L}^{-1} \\{ X(s) \\}$ of a rational Laplace transform $X(s)$ in a systematic way. It is well suited for an algorithmic realization. However, for manual calculus it may be more efficient to classify the poles with respect to their location in the $s$-plane and their symmetries. The classification can then be used to formulate a modified partial fraction decomposition which limits the need for later algebraic simplification of the inverse Laplace transform. Three classes of poles are typically considered\n",
    "\n",
    "| Type | Pole-Zero Diagramm | $X(s)$ | $x(t) = \\mathcal{L}^{-1} \\{ X(s) \\} \\qquad \\qquad$ |\n",
    "|---|:---:|:---:|:---:|\n",
    "| Single complex pole | ![Single pole](single_pole.png) | $\\frac{n!}{(s + s_0)^{n+1}}$ | $t^n e^{-s_0 t} \\epsilon(t)$ |\n",
    "| Conjugated imaginary poles | ![Conjugated imaginary poles](conjugated_imaginary_poles.png) | $\\frac{A s + B}{s^2 + \\omega_0^2}$ | $\\begin{cases} \\sin(\\omega_0 t) \\epsilon(t) \\\\ \\cos(\\omega_0 t) \\epsilon(t) \\end{cases}$ |\n",
    "| Conjugated complex poles | ![](conjugated_complex_poles.png) | $\\frac{A s + B}{(s + \\sigma_0)^2 + \\omega_0^2}$ | $\\begin{cases} e^{-\\sigma_0 t} \\sin(\\omega_0 t) \\epsilon(t) \\\\ e^{-\\sigma_0 t} \\cos(\\omega_0 t) \\epsilon(t) \\end{cases}$ |\n",
    "\n",
    "where $s_0 \\in \\mathbb{C}$ and $\\omega_0, \\sigma_0 \\in \\mathbb{R}$. The expansion coefficients $A, B \\in \\mathbb{R}$ can be derived by comparison of coefficients. Whether $x(t)$ contains a sine or cosine depends on the coefficient $A$. If $A \\neq 0$ then $x(t)$ contains a cosine (cf. [table of selected transforms](table_theorems_transforms.ipynb#Selected-Transforms))."
   ]
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   "metadata": {},
   "source": [
    "#### Example - Inverse transform of a rational Laplace transform with symmetric poles\n",
    "\n",
    "The inverse transform $x(t) = \\mathcal{L}^{-1} \\{ X(s) \\}$ of \n",
    "\n",
    "\\begin{equation}\n",
    "X(s) = \\frac{2 s^2 + 14 s + 124}{s^3 + 8 s^2 + 46 s + 68} \\qquad \\text{for } \\Re \\{s \\} > -2\n",
    "\\end{equation}\n",
    "\n",
    "is computed. First the function $X(s)$ is defined in `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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      "text/latex": [
       "$$\\frac{2 s^{2} + 14 s + 124}{s^{3} + 8 s^{2} + 46 s + 68}$$"
      ],
      "text/plain": [
       "     2               \n",
       "  2⋅s  + 14⋅s + 124  \n",
       "─────────────────────\n",
       " 3      2            \n",
       "s  + 8⋅s  + 46⋅s + 68"
      ]
     },
     "execution_count": 11,
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   ],
   "source": [
    "X = (2*s**2 + 14*s + 124)/(s**3 + 8 * s**2 + 46*s + 68)\n",
    "X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The poles of $X(s)$ are derived by computing the roots of the denominator polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAUYAAAAVBAMAAADIlFouAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEER2zVSJ3ZkiZjKru+/kt61jAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADZElEQVRIDc1WS2sTURT+Jmke0zw6UBcKYoMbXdWILoqIzULpRjD9BY0g4s7URaeoaMQsArZUF5UoSAfciBsDQl0oOCBCUYtPcCFoFKQbpa1aSu0i3sfM3DvJ5DaKoHcxc893vnPON3fuPTMAtO34f8f0ZaZt7/DfkViDxhP+WboEsKU1MjFvEDCyLHmq5XeSlRiVjKbp1ROvmpDFgybJNyCh92j+4KEfn8z4PekfQ6RcuCRQVj5lEyBaFKhu42TGM7eWZfkezCZaCXfyfshcPUqAMQ/U5o6017gZ3Ssek01i58cIPVbwUF6eyZM1Jg2k6x4J4fYawyvoywkmndl+k1iD7TUO5rUfnB+y+D3Jb9KVlW/RmF5Gck2wFBpDn9CTE0w6s/0msRQa+zNwNEayPK5TjckikuuilEIjIY1Ygklntt8klkIjOQc/Of93NZKo2BINHWfxSo2h04wjLk/2XII2wFYlVmKwUuMOi4d6Gm8O1NB7kYFSefau4znOda4PaRFnr6g0JvY/9cUBLzGTiYbqFHW2qkrj3Wc1Hu9qTBn6NxT6KSiXD9cJMGORixj8EQwGqDSSd50XUXyWKk7zJ9a4rdKIiHOuXY0k5K2eL7NIqbxOaA9OEVQbMukgTxYvMRK/BGgUTJADBmxigWc4P75m9GSl+ACNEn+RKImb5rkF07R50NQVfJfCefkbh6CVuWjHd0viqHpPooaUv8GlcoivY0TOFqDRTX8f+JBnhruOb4Cpmu+DwjUeywCskbuhkQKuuXPS9Nv3x77lZo1p0hOWcFhEq8611jCaNS4Az9FlS/GsfIT2QrmHYxbYLViU5GwtAfJZ2ubvWuBRIP1CWyE3b9B1lNfVcwBfAPquyXDXsQB8RV+e3NzBNEbrxJQ16uOViSLBpIPV/DVxEnRbzf1RL+GAEVq6TgniXLtfEyfMu91u6Y9VhD9il24RilSey5M1djUajRzNw0538v3qI8RGqd06dlaa/ymqExeA1xal8v44v3g2C/qfETBClckMh9111CtzBlL7GCjKt2oMSIbhILBzLBKsUSRwNQrEN2NLKK+jz+sYhSCwc0zen4FR7j9FoNPZiqyRtyEQWMu293Xi6e2EpOCwj4L2WcEAupXejZ3WxhQlYxtLUH2sJP1b5ywR9ws1ccwXxFD59wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ -2 : 1, \\quad -3 - 5 i : 1, \\quad -3 + 5 i : 1\\right \\}$$"
      ],
      "text/plain": [
       "{-2: 1, -3 - 5⋅ⅈ: 1, -3 + 5⋅ⅈ: 1}"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poles = sym.roots(sym.denom(X))\n",
    "poles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result is a real-valued pole and a conjugate complex pair of poles. According to above introduced classification of poles, the following ansatz is chosen for the partial fraction decomposition of the Laplace transform\n",
    "\n",
    "\\begin{equation}\n",
    "X_p(s) = \\frac{A}{s + 2} + \\frac{B s + C}{s^2 + 6s + 34}\n",
    "\\end{equation}\n",
    "\n",
    "The coefficients $A, B, C \\in \\mathbb{R}$ are derived by equating coefficients with $X(s)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ A : 4, \\quad B : -2, \\quad C : -6\\right \\}$$"
      ],
      "text/plain": [
       "{A: 4, B: -2, C: -6}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A, B, C = sym.symbols('A B C', real=True)\n",
    "\n",
    "Xp = A / (s+2) + (B*s + C)/(s**2 + 6*s + 34)\n",
    "coeffs = sym.solve(sym.Eq(X, Xp), (A, B, C))\n",
    "coeffs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Introducing the coefficients into $X_p(s)$ yields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{- 2 s - 6}{s^{2} + 6 s + 34} + \\frac{4}{s + 2}$$"
      ],
      "text/plain": [
       "   -2⋅s - 6       4  \n",
       "───────────── + ─────\n",
       " 2              s + 2\n",
       "s  + 6⋅s + 34        "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Xp = Xp.subs(coeffs)\n",
    "Xp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first fraction belongs to the complex conjugate poles. Applying [completion of the square](https://en.wikipedia.org/wiki/Completing_the_square) to the denominator, its inverse can be identified in the [table of Laplace transforms](table_theorems_transforms.ipynb#Transforms) as exponentially decaying cosine signal. Performing the inverse Laplace transform with `SymPy` yields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 2 e^{- 3 t} \\cos{\\left (5 t \\right )} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "    -3⋅t                      \n",
       "-2⋅ℯ    ⋅cos(5⋅t)⋅Heaviside(t)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x1 = sym.inverse_laplace_transform(Xp.args[1], s, t)\n",
    "x1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The second fraction belongs to a real-valued pole of first degree. Its inverse Laplace transform can be looked-up directly in the [table of Laplace transforms](table_theorems_transforms.ipynb#Transforms) as exponentially decaying signal. Performing the inverse Laplace transform again with `SymPy` yields"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$4 e^{- 2 t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "   -2⋅t             \n",
       "4⋅ℯ    ⋅Heaviside(t)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x2 = sym.inverse_laplace_transform(Xp.args[0], s, t)\n",
    "x2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inverse Laplace transform of $X(s)$ is given by summing up these two parts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$4 e^{- 2 t} \\theta\\left(t\\right) - 2 e^{- 3 t} \\cos{\\left (5 t \\right )} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "   -2⋅t                   -3⋅t                      \n",
       "4⋅ℯ    ⋅Heaviside(t) - 2⋅ℯ    ⋅cos(5⋅t)⋅Heaviside(t)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = x1 + x2\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The resulting signal is plotted for illustration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": 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       "<Figure size 614.4x460.8 with 1 Axes>"
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    }
   ],
   "source": [
    "sym.plot(x, (t, -0.1, 4), xlabel='$t$', ylabel='$x(t)$');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The same result can be derived directly from $X(s)$ by using the inverse Laplace transform of `SymPy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$2 \\left(2 e^{t} - \\cos{\\left (5 t \\right )}\\right) e^{- 3 t} \\theta\\left(t\\right)$$"
      ],
      "text/plain": [
       "  ⎛   t           ⎞  -3⋅t             \n",
       "2⋅⎝2⋅ℯ  - cos(5⋅t)⎠⋅ℯ    ⋅Heaviside(t)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.inverse_laplace_transform(X, s, t).simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Derive the inverse Laplace transform of $X(s)$ by manual calculation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
   ]
  }
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